Optimal Cytoplasmic Transport in Viral Infections
Maria R. D’Orsogna1, Tom Chou2*
1 Department of Mathematics, California State University Northridge, Los Angeles, California, United States of America, 2 Department of Biomathematics and Department
of Mathematics, University of California Los Angeles, Los Angeles, California, United States of America
Abstract
For many viruses, the ability to infect eukaryotic cells depends on their transport through the cytoplasm and across the
nuclear membrane of the host cell. During this journey, viral contents are biochemically processed into complexes capable
of both nuclear penetration and genomic integration. We develop a stochastic model of viral entry that incorporates all
relevant aspects of transport, including convection along microtubules, biochemical conversion, degradation, and nuclear
entry. Analysis of the nuclear infection probabilities in terms of the transport velocity, degradation, and biochemical
conversion rates shows how certain values of key parameters can maximize the nuclear entry probability of the viral
material. The existence of such ‘‘optimal’’ infection scenarios depends on the details of the biochemical conversion process
and implies potentially counterintuitive effects in viral infection, suggesting new avenues for antiviral treatment. Such
optimal parameter values provide a plausible transport-based explanation of the action of restriction factors and of
experimentally observed optimal capsid stability. Finally, we propose a new interpretation of how genetic mutations
unrelated to the mechanism of drug action may nonetheless confer novel types of overall drug resistance.
Citation: D’Orsogna MR, Chou T (2009) Optimal Cytoplasmic Transport in Viral Infections. PLoS ONE 4(12): e8165. doi:10.1371/journal.pone.0008165
Editor: Laurent Rénia, BMSI-A*STAR, Singapore
Received September 3, 2009; Accepted November 10, 2009; Published December 30, 2009
Copyright: ß 2009 D’Orsogna, Chou. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported by the Division of Mathematical Sciences of the National Science Foundation through grants DMS-0719462 (MRD) and DMS0349195 (TC), and by the National Institutes of Health through grant K25AI058672 (TC). Part of this work was performed during the ‘‘Optimal Transport’ program
at the Institute for Pure and Applied Mathematics (IPAM) at University of California, Los Angeles. The funders had no role in study design, data collection and
analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: [email protected]
[5–9]. Although the exact mechanisms are unknown, during its
journey, the RTC sheds some of its proteins and acquires others.
Ultimately, a preintegration complex (PIC), capable of nuclear
entry, is formed [5,7,9].
Two key features of nuclear entry are thus: i) the microtubule
assisted directional motion, and ii) the sequence of transformations
from capsid to preintegration complex. Recent experiments using
fluorescence imaging have revealed that cytoplasmic transport and
viral transformation occur along microtubules, assisted by
microtubule-associated molecular motors of the host cell [12–15].
Theoretical studies of viral transport in the cytoplasm include that
of Lagache et al. [16], who studied viral transport as a microtubulemediated convection, punctuated by free cytoplasmic diffusion. In
their work, the motion of a single viral particle can be reduced to
an effective drift, where the normalized velocity depends on
molecular motor speeds, microtubule density, and viral detachment and attachment rates on the microtubule.
Biochemical transformations of the viral material are known to
also play a critical role in determining the probability and timing
of productive infections. Experiments that enhance or hinder
specific viral transformations in the cytoplasm show that artificial
delays or accelerations can result in abortive infections [9,17–22].
In particular, various TRIM5 a proteins that accelerate capsid
disassembly appear to inhibit infection, providing e.g., protection for
humans against SIV [17,18]. Entry failure may be due to
increased cytoplasmic decay of the viral material, to biochemical
constraints blocking nuclear entry, or to biochemical deficiencies
that prevent integration with the host DNA. The emerging
consensus is that biochemical transformations and transport must
be balanced before successful infection can occur [23,24].
Introduction
In order to reproduce, viruses must exploit the internal
machinery of host cells to synthesize key proteins and assemble
new virions. The genetic material of membrane-enveloped viruses
is contained within an internal protein capsid enclosed by a lipid
membrane. Upon contact with a cell, complex interactions
between cellular surface receptors and viral spike proteins [1,2]
induce fusion between viral and host cell membranes, allowing the
protein capsid to enter the cell cytoplasm. For many viral species,
the genome must also penetrate the nucleus and integrate with the
host DNA. Some viruses wait for dissolution of the cell nuclear
membrane during mitosis for genomic integration; others take a
more active approach by directly transferring their RNA or DNA
through nuclear pores. This infection mechanism allows viral
reproduction at any stage of the cell cycle, and is utilized by
lentiviruses such as HIV [3–5].
Getting to the nucleus from the cell periphery is a treacherous
journey since viruses must navigate the cytoplasm, a crowded
environment where diffusion is inhibited [6–8] and degradation
may take place [9,10]. Moreover, a series of biochemical steps,
such as capsid disassembly and reverse transcription, must occur
so that viral material is transformed into complexes that are able to
enter the nucleus and integrate with the host genome.
The post-entry dynamics of HIV is especially complex since
HIV capsids rapidly disassemble and give rise to intermediate
structures that are difficult to visualize, even with modern
fluorescent probe tracking techniques [11]. One of the capsid
derivatives is the reverse transcription complex (RTC), through
which RNA is processed into DNA while traveling to the nucleus
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However, there is no theoretical model showing how acceleration
of transformations might reduce viral infection. Furthermore,
although several authors have considered viral trafficking and
microtubular transport [16,25–27], the coupling between transport and biochemical transformations of viral material, and how
this can enhance or suppress infection probabilities has remained
largely unexplored.
In this paper, we tackle these questions through a stochastic
model where viral entry depends on the interplay between physical
transport along microtubules and the required serial biochemical
transformations of the viral material, as observed in numerous
studies [7–9]. The mathematical model relies on many simplifying
assumptions, but captures the main steps involved in nuclear entry.
One important consequence of this cytoplasmic transport model of
viral infection is the emergence of optimal values of key
parameters that the maximize nuclear entry probability, and
consequently the productive infection efficiency. We analyze the
probability and timing of nuclear entry, providing a possible
explanation and quantitative framework for recent discoveries of
an optimal capsid stability for HIV infections [17,19,20,23]. Our
findings also suggest a broader mechanistic interpretation of
antiviral drug efficacy and resistance.
Figure 1. Schematic of one-dimensional cytoplasmic transport
along a microtubule. Virus capsids, after entering the cell and
passing through the actin rich cortical region, attach to molecular
motors at the distal end of the microtubules at x~0. Time is defined
with t~0 corresponding to the moment the virus capsid escapes the
actin layer and attaches to a microtubule-associated motor. The viral
material is biochemically processed through a chain of intermediate
states n with rate cn , while simultaneously carried by microtubuleassociated molecular motors moving at velocity vn towards the nucleus
at x~L. During transport, the virus particle degrades with total rate mn .
After microtubule motor assisted transport, the viral material is
deposited into a perinuclear (PN) region of thickness ‘%L, from which
it enters the nucleus with rate kn . Only viral particles that enter the
nucleus in state n lead to a productive infection.
doi:10.1371/journal.pone.0008165.g001
Methods
Here, we develop a stochastic model describing the transport,
transformation, and degradation of viral material in the host cell
cytoplasm. Once inside the cell, the capsid of a newly fused
enveloped virus encounters a dense actin-rich cortical region that
hinders its diffusion [8]. Transport to the nucleus is mediated by
microtubules that penetrate this actin-rich layer, creating a highway
through the cytoplasm. We thus consider an effective onedimensional viral motion along microtubule tracks that extend
from the cell periphery at x~0 to the perinuclear (PN) region near
x~L, as shown in Fig. 1. The capsid is assumed to bind to
microtubule-associated motors (such as dynein) at x~0 before it is
convected towards the microtubule organizing center, near the
perinuclear (PN) region. Processed viral material is then deposited in
the thin PN layer before being transported across nuclear pores [3].
We focus on viral dynamics after attachment to the microtubule-associated motor. Viral material initially at x~0 undergoes a
series of transformations among n long-lived, distinguishable
states while simultaneously being convected by motors with
velocity vn . Starting from a specified initial state denoted by
n~0, a series of irreversible transformations, such as capsid
disassembly or reverse transcription, take place from state n to
state nz1 until the infective state n is reached. We assume that
only viral material entering the nucleus in state n can lead to
incorporation into the host genome and productively infect the
cell. Since transitions are assumed to be sequential, the initial state
n~0 may correspond to any viral intermediate that can be
quantitatively detected. For example, the ‘‘initial’’ state at time
zero in our Markov chain might correspond to the state where
minus-strand DNA syntehsis has just been completed within the
reverse transcription process.
The time evolution for the probability density Mn (x, t) of
finding microtubule-associated viral particles in state n, between
positions x and xz dx, at time t, is described by the transport
equation
LMn
LMn
zvn
~cn{1 Mn{1 {(cn zmn )Mn ,
Lt
Lx
nz1, with c{1 ~0. For many viruses such as HIV, fusion and
initial entry are rate limiting steps and the density of virus particles
in the cytoplasm is low, allowing us to consider independent
particles. Furthermore, a typical dynein motor has high processivity and a run length of approximately 2 m m [28], which is
comparable to the length of the microtubules spanning the radius
of a typical T-cell. Since backtracking and detachment/reattachment of high-processivity motors is rare, we model the viral
transport as a purely convective process, with an effective velocity
vn representing an average over the intrinsic molecular motor
velocity and the zero drift velocity when the virus-motor complex
is occasionally stalled or randomly diffusing in the cytoplasm.
However, detachment of the reverse transcriptase (RT) machinery
from the complex and general degradation, represented by mn , are
considered irreversible. Since further transformations beyond state
n do not lead to productive infections and are equivalent to
degradation, we can also set cn ~0 but use an appropriate
degradation mn . Our initial condition is Mn (x, t~0)~d(x)dn,0 .
The boundary condition at x~L is derived by introducing a PN
layer of thickness ‘%L at the end of the microtubule where motors
unload their viral cargoes [3,14]. Within this thin layer, Pn (t), the
density of viral material in state n, is assumed uniform so that the
total amount of viral material in the PN layer, ‘Pn (t), obeys
‘
ð2Þ
n and cn are the degradation and transformation rates
Here, m
specifically within the PN region. The infection probability also
depends on the nuclear import rate kn , which is a function of the
biochemical composition of the nth intermediate, the nuclear pore
density and structure [16], and possibly chaperones that actively
transport material into the nucleus [30]. Although perinuclear
ð1Þ
where cn is the rate of irreversible transition from state n to state
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dPn
~vn Mn (L,t){‘(kn z
mn zcn )Pn z‘cn{1 Pn{1 :
dt
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Cytoplasmic Viral Transport
material at stage n=n may also enter the nucleus, incompletely
processed genetic material is usually not viable for integration with
the host DNA [9,22]; nuclear entry of premature intermediates
can be considered to be equivalent to degradation in the PN
region. Both probability densities, Pn (t) and Mn (t) have units of
probability per unit length and all of our results will be
independent of ‘ in the ‘=L?0 limit.
Eqs. 1–2, along with the initial condition Mn (x, 0), are the basic
equations of our model. Since the productive infection probability
will be proportional to the nuclear entry probability, we construct
the probability Q that nuclear entry eventually occurs by timeintegrating the flux ‘kn Pn (t) of the intermediate n into the
nucleus:
ð?
ð3Þ
Pn (t’)dt’:
Q~‘kn
and convected by a single molecular motor. We will assume that a
shedding virus particle will not sufficiently change its drag as to
affect its transport velocity. At typical convection speeds on the
order of *0:1{1mm/sec, an intact viral capsid of diameter
d*50 nm in cytoplasmic fluid of viscosity g*5|10{2 Pa imparts
a drag force Fd *20 fN which is a negligible fraction (*0:1%) of
the typical stall force of the dynein motor. Besides changing
changing hydrodynamic size, different viral states n may carry
inherently different motor detachment rates, changing the ratio of
convective to diffusive transport. Therefore, it is possible that
different states n are described by different effective transport
velocities vn . However, as long as the motor processivities are high,
and convection along microtubules is more prevalent than
cytoplasmic diffusion, the effective velocity can be approximated
as state-independent. The only possibility of a state-dependent
velocity vn , which we neglect for the sake of mathematical
simplicity, is if motors of different velocity or stalling frequency are
interchanged during the viral maturation process.
Table 1 below summarizes all variables and parameters defined
in our model. Throughout our study, time and length units are
seconds and microns, respectively. Bounds for the parameter
ranges are estimated from the literature and from physical
considerations. Since the definition of the states n is widely
varying, from different levels of capsid disassembly, to different
stages of the reverse transcription process, the parameters are
effective rates that can represent different processes. For example,
even though motor velocities are *0:1{0:6mm/s, recent
experiments imply that RT detaches and reattaches a number of
times during the entire reverse transcription process [29]. In this
scenario, both the effective transformtion rates c and velocity v
associated with reverse transcription may be significantly reduced
from the nucleotide addition rate and the motor velocity,
respectively. Nonetheless, we will demonstrate (cf. Supporting
Information) that the qualitative features of our model arise for a
very wide range of parameters and is insensitive to details.
The entire infectivity process includes not only transport to the
nucleus but also the subsequent stages of DNA integration and
daughter virion assembly. Since the total infection probability is a
product of the efficiencies of each of these processes, it will be
proportional to the nuclear entry probability Q that we calculate,
and to the other post nuclear penetration events that also
0
Ð
~ n (s)~ ? Pn (t)e{st dt, we
Upon defining the Laplace transform P
0
~ n (s~0). To
can express the nuclear entry probability as Q~‘kn P
solve for Q, we first take the Laplace transform of Eq. 1 and solve
~ n (x, s):
for M
~ n (x, s)~ c0 . . . cn{1
M
v0 v1 . . . vn
n
X
e{Rj (s)x
,
P nk=j (Rk (s){Rj (s))
j~0
n§0, ð4Þ
where vn Rn (s)~(szmn zcn ) for n=n and vn Rn (s)~(szmn ).
Equation 4 represents an explicit expression for the Laplacetransformed nth - state viral particle probability density at position
~ n (x, s) into the Laplacex in the cytoplasm. Substitution of M
~ n (s) and hence Q:
transform of Eq. 2 allows us to solve for P
Q~
n
ci
kn X
~ j (L, s~0) P n {1
vj M
:
i~j
kn z
mn j~0
ki zci z
mi
ð5Þ
Each term of the sum in Eq. 5 represents the probability of arrival
of the virus particle to the PN region in state jƒn , and its
subsequent conversion to state n before being able to enter the
nucleus. Equation 5 is our main mathematical result.
Because the exact biochemical fate of viral material within the
PN layer is not well established, we will explore the consequences
of our model in two limits. Since the PN region can be dense with
actin meshwork, and is not traversed by microtubules, degradation
and transformation rates may be negligible compared to those in
the cytoplasm. This may occur, for instance, because relevant
proteins such as proteasomes are too large to penetrate the dense
actin meshwork near membrane-bound compartments such as the
nucleus [31]. In this case cn ~0 in Eq. 2, and the nuclear entry
~ n (L, 0)=(kn zmn ),
probability is simply Q:Qinert ~kn vn M
independent of kn=n since only material that reaches the PN
region at state n can enter the nucleus. In the opposite limit of a
perinculear layer where all transformations and degradations rates
n ~mn , and the nuclear entry probability
are unhindered, cn ~cn , m
is denoted Q:Qactive .
Our equation for Q can easily be modified to include all viral
states m that are not microtubule bound and have vm ~0. In such
cases, Eq. 5 for Q P
is modified
by summing terms only over those
states that move ( nj~0,=m . . .) and by multiplying the resulting
cm
expression by the probability
that the virus material
mm zcm
survives each of the nonmoving states m.
In our subsequent analysis of Eq. 5, we will assume that all states
n are transported velocities vn ~v if they are microtubule-bound
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Table 1. Variable and parameter definitions.
Variable or
parameter
Physical definition
Units and typical
values
0ƒnƒn
state of the viral material
n &2{5
v
effective velocity
0.01–0.6 mm sec{1
cn , cn
transformation rates
10{3 {0:5 sec{1
n
mn , m
decay rates
10{5 {10{2 sec{1
kn
nuclear entry rates
10{3 sec{1
L
cytoplasmic distance
*4mm
‘
PN layer thickness
0.05 mm
Pn (t)
probability density in PN layer
length{1
Mn (x, t)
probability density at position x
length{1
Qinert
entry probability, inert PN
(
mn ~cn ~0)
0ƒQinert ƒ1
Qactive
entry probability, active PN
0ƒQactive ƒ1
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active case, immature perinuclear material at stages n=n can
transform to state n within the PN layer, enter nuclear pores, and
add to Q. This is not possible in the inert case where perinuclear
material at stage n=n cannot contribute to infection.
As expected, Eq. 7 predicts that Qinert monotonically decreases
with increasing m1 ,m0 and increases with k1 . However, Qinert can
be a non-monotonic function of the initial conversion rate c0 ,
depending on values of the other parameters, as indicated in
Fig. 2(a). If m0 §m1 , Qinert is a monotonically increasing function of
c0 , while if m0 vm1 , Qinert can have a maximum as a function c0 .
This result can be understood physically. If m0 vm1 , survival would
be increased by decreasing the conversion rate c0 from state n~0
to state n ~1 allowing a longer life-span in the less degradative
state n~0. However, viral material must eventually convert to
state n ~1 within the microtubule travel time L=v for the
infection to be productive. These opposing constraints for the
conversion rate lead to an intermediate value of c0 which
maximizes Qinert . This maximum does not arise when m0 §m1
because there is no survival benefit for viral material to stay in the
unprocessed stage at n~0. Experimental evidence consistent with
local maxima in Q arises, for example, in the protein TRIM5 a
which has been shown to restrict viral infection by accelerating the
uncoating of retroviral capsids (increasing, say, c0 ) [17,18].
The behaviors described above can be generalized to larger n .
Regardless of the number n of intermediates, Qinert is a
decreasing function of mn and an increasing function of kn . We
can explore the dependence of Q on transformation rates cn under
a wide range of scenarios by considering various decay/
degradation rate sequences fmn g. As above, for simplicity we
assume that degradation rates within the PN region are the same
n ~mn . Figure 3 shows the entry probabilities
as in the cytoplasm: m
for n ~2 corresponding to four qualitatively different sequences
fmn g that: monotonically increase, contain a maximum, contain a
minimum, and monotonically decrease.
For monotonically increasing mn (mnz1 wmn ), optimal values of
the conversion rates cn that yield a local maximum in Qinert arise. If
contribute to replication efficiency losses. Finally, it is important to
realize that our model of n intermediates can be applied to any
experimentally probed subset of all the intermediate steps of the
entire, sequential infection process. Therefore, the state n~0 can
correspond to any initially observed state if one is interested in the
nuclear entry probability conditioned on starting from the
specified intermediate.
Results and Discussion
Eqs. 5 and 4 show that Q always decreases with increasing
degradation rate mn . The entry probability Q also increases with
kn and decreases with kn=n . However, for various fixed decay
rate patterns fmn g, Q can depend on the transformation rates cn in
unexpected ways. Therefore, we first explore in detail how Q
depends on the transformation rates cn .
The essence of our model is captured by considering the kinetics
of a small number of longest-lived intermediate states that control
the entry probability Q. For example, n &3 may correspond to
the early, middle, and late stages of reverse transcription in HIV,
as detectable by quantitative PCR [32]. For simplicity, we first
assume only one transformation step, n ~1, is required for
productive infection. The entry probability from Eq. 5 is
Q~Qinert z
k1c0
v
~ 0 (L, 0),
M
k1 z
m1 k0 zc0 z
m0
ð6Þ
where
(m0 zc0 )
Qinert ~
m1
k1 c0 e{ v L {e{ v L
(k1 z
m1 ) m1 {(m0 zc0 )
ð7Þ
is the entry probability when perinuclear transformations do not
1 ~m1 . Mathematoccur (cn ~0). For simplicity, we assume that m
ically, QwQinert ; physically, this inequality arises because in the
Figure 2. Entry probability Qinert for viruses with a single intermediate step (n ~1). Qinert is shown as a function of c0 for different values of
m1 . Parameters are v~0:2mm/s, L~4mm, kn ~0:001s{1 . Henceforth, all parameters will be defined with time and length expressed in units of
m0 ,m1 ~
seconds and microns, respectively. (a) Optimal conversion rates exist only if m0 ƒm1 , as shown by the solid curve for m0 ~0:01 and m1 ~0:49. When
m1 ~m0 ~0:25 (circles), Qinert saturates as a function of c0 , while for m0 wm1 (dashed) Qinert increases monotonically. When m1 {m0 ~2v=L~0:1, Qinert is
maximized exactly at c0 ~2v=L~0:1. Panel (b) shows Qinert as a function of c0 and m0 for m1 ~0:1. Values of c0 that yield local maxima in Qinert exist
only if m0 v0:1.
doi:10.1371/journal.pone.0008165.g002
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Figure 3. Entry probability Q as a function of ª0 and ª1 forPn ~2. The motor velocity, microtubule length, and nuclear import rate are set to
v~0:2,L~4 and kn ~0:001: Four sets of fm0 , m1 , m2 g, with fixed 2i~0 mi ~0:098, are explored. For (a–d), an inert PN region is considered (cn ~0). (a)
When m0 vm1 vm2 (fm0 , m1 , m2 g~f0:004, 0:014, 0:08g), a local maximum in Qinert arises. (b) When the interior degradation rate m1 is largest, a
maximum in Q as a function of c0 arises for large c1 . (c) When the initial and final states have larger degradation rates, a maximum as a function of c1
arises for large c0 . (d) For decreasing degradation rates (fm0 , m1 , m2 g~f0:06, 0:024, 0:014g), Qinert monotonically increases in both c0 and c1 . The
n ~mn ) are shown in (e–h). The maxima that arise are generally sharper
analogous infection probabilities Qactive for active PN regions (cn ~cn and m
here than in the corresponding inert cases. The hypothetical strains depicted by (f) and (g) have different degradation phenotypes fmn g, but respond
differently to a c1 -reducing antiviral. In (f), Q decreases from Q&0:017 at point A (c0 ~c1 ~0:8) to Q&0:011 at point B where c1 has been reduced to
c1 ~0:05. However, in strain (g), reducing c1 by the same amount increases Q from 0.017 to 0.021. Only at points C, when c1 &0 is Q&0 and infection
suppressed.
doi:10.1371/journal.pone.0008165.g003
the sequence of mn rates is not monotonically ordered, then saddle
points, or maxima in Q as a function of one of the cn , will generally
arise. These are illustrated in Fig. 3(a–d) for n ~2, under various
degradation sequences that sum to 0:098. Only in the monotonically decreasing case mnz1 vmn is Q monotonic in all cn , and does
not exhibit a local maximum. As discussed earlier, when considering
a biochemically active PN region, we consider the case cn ~cn in
Eq. 2. When cn w0, additional terms arise in Eq. 5 that increase the
total infectivity Q. Figs. 3(e–h) show Q at the same parameter values
used to determine Qinert in Figs. 3(a–d), except that cn ~cn and
n ~mn . The presence of an active layer yields much sharper and
m
more intense maxima in Q as a function of the transformation rates
c0 ,c1 . Figs. 3(a), (b) and (e) show infection probabilities that can
decrease with increasing c0 . Such phenomena has been observed
when TRIM5 a activity is increased [17,18], and in nucleocapsid
mutations that cause premature reverse transcription [20]. Both of
these biological examples increase the initial transformation rates,
yet decrease overall infectivity. Furthermore, mutational studies
have shown that an optimal capsid stability is also required for
maximal reverse transcription and productive infection
[17,19,20,23]. This suggests an optimal value also arises for the
transformation rate associated with capsid disassembly, which is
exhibited by the mathematical model.
Although any of the four pictured degradation sequences, or
more complicated ones for larger n , are potentially realizable,
there is direct evidence for a maximum in the mn sequence for
intermediate states of reverse transcription. In particular, Thomas,
Ott, and Gorelick [10] measured a high-low-high pattern of stepwise efficiencies for early, middle, and late stages of reverse
transcription. This pattern corresponds to a low-high-low pattern
for the combination of parameters mn =(mn zcn ). In Figs. 3(b) and
(f), we assumed the correspondingly low-high-low structure for mn
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estimated from measured transformation efficiencies [10] and
show how Qinert and Qactive behave as c0 , c1 are varied.
In light of this comparison, the structure of our infection
probability Q suggests an alternative conceptual guide to the
development and administration of certain classes of anti-retroviral
drugs such as reverse transcriptase (RT) inhibitors that act to
reduce rates of subsequent transformations (lowering say, c1 ). If
viral complexes become less susceptible to degradation as they
progress towards the nuclear entry competent stage such that
mn §mnz1 , then viral infectivity will be decreased upon lowering
any transformation rate cn such as by the administration of the RT
inhibitor, as can be seen from Fig. 3(d) and (h). On the other hand,
if viral intermediate states are more susceptible to degradation
exist such
such that mn vmnz1 , optimal theoretical values of cmax
n
that Q is maximized. Intrinsic transformation rates cn may not
necessarily be near the theoretical optimal values cmax
n .
The complex parameter dependence of the infection probability
Q also suggests a subtle interpretation of ‘‘hidden’’ mutations that
confer drug resistance to antivirals such as RT inhibitors [33]. In
our model, genetic mutations that induce molecular changes
unrelated to those that affect drug binding may still impart a more
global, transport-dependent drug resistance. Specific mutations that
change the transport properties (in addition to changing c) are
known to exist and are also known to interact with each other in
unexpected ways. For example, the M184V and L74V mutations
are known to decrease RT processivity by increasing RT
detachment [34,35], which is equivalent to increasing specific mn
in our model. Qualitatively, a mutational change in mn can be
described by, for example, a shift of the entry probability from that
shown in Fig. 3(f) to the one in Fig. 3(g). The administration of an
antiviral that hinders RT on a ‘‘wild-type’’ viral strain (f), may be
represented by A?B. A drug-resistant mutation that affects the
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Cytoplasmic Viral Transport
sensitive to perinuclear transformation and nuclear entry rates. In
the case of an active PN layer where only material at state n can
enter the nucleus (cn =0 and kn=n ~0), the entry probability Q,
although greater than Qinert , becomes a monotonically increasing
function of v and loses its maximum as shown by the dashed curve in
Fig. 4. In this case the under-processed viral material simply waits in
the PN layer, eventually converting to state n , whereupon nuclear
entry is possible. Thus, there is no penalty in reaching the PN region
in state n=n , and the entry probability Q increases with v since
higher velocities allow the virus to better escape cytoplasmic
degradation with the possibility of completing biochemical processes
upon arrival at x~L.
Now, suppose that viral material arriving in the PN region at
state n=n can enter the nucleus at rate kn=n , but still without
leading to productive infection. The entry of this improperly
processed material does not contribute to Q. All else being equal,
allowing the nuclear entry of incomplete material yields a smaller
entry probability Q compared to the previous case of an active
perinucleus with kn=n ~0. Here, the maximum in Q as a function
of v is recovered, as shown by the solid black intermediate curve in
Fig. 4. As in the case of an inert PN layer, a maximizing transport
velocity v arises, albeit for different biochemical reasons. While
higher transport velocities v allow for less degradation in the
cytoplasm, reaching the PN region in a state nvn increases the
probability of unproductive nuclear import.
Our model illustrates how infection probabilities can be
controlled by varying motor velocities, by adjusting e.g., cellular
ATP concentration. Although varying ATP concentrations may
also affect both the degradation and transformation rates of the
infection process, it does suggest in principle that varying motor
velocities might be used to probe viral kinetics within the PN
region. If higher velocities v always lead to higher measured
infection probabilities, the PN region is active and only completely
processed material can enter the nucleus. As shown by the dashed
curve in Fig. 4, an increasing Q as a function of v corresponds to
the case of cn =0 and kn=n ~0. On the other hand, if a maximum
is observed as a function of v, the PN region is either inert (cn ~0),
or is active but facilitates nuclear entry of incompletely processed
viral material (cn =0, kn=n =0). To distinguish between these two
cases we propose in the Supporting Information, another
independent measurement: the mean first time of productive
infection, conditioned on realization of a productive infection (cf.
Fig. 1 of File S1). However, the qualitative features of the
computed nuclear entry probability, Q, are fairly robust with
respect to interdependences of our parameters. For example, Fig. 2
of File S1 shows that qualitatively reasonable velocity and
transformation rate dependences on ATP concentration can lead
to optimal entry probabilities at intermediate ATP concentrations.
Finally, we note that all results depend on transport velocity and
cell radius through the combination L=v. Therefore, the
qualitative dependences on v also hold for 1=L; specifically, for
a fixed transport velocity, we expect that for a host cell with an
inactive perinucleus, a particular host cell radius L that optimizes
nuclear entry probability may arise.
The rate parameters in Table 1 and used to generate our curves
are very rough estimates inferred from the literature on infectivity
assays. Recent experiments show that many of these rates vary
significantly depending on cell type, cell state, and on assay
preparation [15]. As a result, the above values are indicative and
may change depending on the particular HIV sample under
investigation. For example, initial delays up to the order of hours
may occur in the peripheral cytoskeleton, adding to an offset in the
zero of time at which the viral material starts to be transported by
microtubules. Prolonged delays have been measured between the
degradation rates alone (through, e.g., decreased RT processivity)
would change the fmn g scenario such that the system would shift
from point B to point B9 in Fig. 3(g). Cessation of drug treatment
would then bring the system to point A9, which has a lower entry
probability that the original, untreated ‘‘wild-type’’ strain at point A.
Many mutations also have complex interactions that in certain
combinations, can resensitize the virus, recovering some amount of
drug susceptibility. For example, the L74V and K65R mutation
together can resensitize HIV-1 to the NRTI zidovudine (AZT) [34].
Many thymidine analog mutations (TAMs) can also ‘‘interact’’ with
the M184V mutation to increase susceptibility to AZT, stavudine,
and tenofovir [36,37]. Such complex mutation interactions may be
explored by considering variations in the fmn , cn g parameter space
of our transport-based model. For example, if a drug-resistant strain
is qualitatively described by the nuclear entry function depicted in
Fig. 3(g), and an additional mutation decreases the RT processivity
(such as the M184V or L74V mutations [34,35]) the virus is now
resensitized to drugs that decrease c1 .
In addition to the rich infection behavior possible under
variations in mn and cn , we identify another key parameter v, the
speed at which viral material is transported along the microtubules.
Dynein motors, which transport viral cargo to the nucleus, can
move with a range of velocities v&0:1{0:6mm/sec, with higher
ATP concentrations leading to higher motor velocities [38,39]. An
analysis of Qinert as a function of v shows that when n ~1, Qinert is
maximized at vmax (n ~1)~(m0 zc0 {m1 )=½ln (m0 zc0 ){ ln (m1 ).
For n w1 optimal speeds that maximize Qinert as a function of v
also arise and can be found numerically; entry probabilities as a
function of v are shown in Fig. 4 for n ~4. The existence of optimal
transport velocities for Qinert results from two opposing effects: high
velocities do not provide sufficient time for the virus particles to
reach the infective state n , while low velocities increase exposure to
cytoplasmic degradation. The behavior of Q as a function of v is also
Figure 4. Entry probabilities as a function of microtubule transport
velocity v. The entry probability Qinert associated with an inert PN region
(circles) is low, but exhibits a maximum. When perinuclear transformations
occur (dashed curve) the virus material converts to infection-competent
species n while it is waiting in the PN region, increasing Q. However, the
maximum in Qactive (v) disappears. When nuclear import of infectionincompetent species are also allowed (kn w0), effective degradation
increases through unproductive entries, and the maximum in
Qactive (v) can reappear. These curve were generated using representative parameters fm0 , m1 , m2 , m3 , m4 g~f0:006, 0:004, 0:0001, 0:01, 0:003g,
fc0 , c1 , c2 , c3 g~f0:5, 0:02, 0:01, 0:03g, k4 ~0:0001, and knv4 ~0:001.
doi:10.1371/journal.pone.0008165.g004
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Cytoplasmic Viral Transport
completion of RT and nuclear entry [15]. These delays appear to
vary in certain cell types, and are incorporated in our model
through the nuclear entry rate kn . However, our qualitative
finding of maxima in nuclear entry probability Q is insensitive to
the magnitudes of kn . We also demonstrate in the Supporting
Information how more complicated parameter interdependencies
may preserve the mathematical structure and qualitative features
of our results. Specifically, we show that if RT transformation rates
c and the transport velocity v are both linearly dependent on ATP
concentration, a maximum nuclear entry probability arises at
intermediate values of ATP.
Summarizing, the cellular physics accompanying viral infections
involves a number of complex and orchestrated biochemical steps.
We have proposed a phenomenological, yet illustrative stochastic
model that incorporates the basic, known processes of viral
transport, degradation, conversion, and nuclear entry in the
infection process. Including only these processes in our model, we
find a rich dependence of the productive infection probability and
the mean time to infection on cytoplasmic and perinuclear
transport parameters.
In vivo, we expect transport rates and perinuclear properties to
be evolutionarily selected according to constraints imposed by the
host cell to optimize a combination of infection speed and infection
probability. We explored and found parameter regimes where
viral infectivity can be optimized under constraints of active or
inactive PN regions and of the nuclear import kinetics.
Mathematically, the infection probability Q, regardless of whether
the PN region is inert or active, is
N
N
N
N
since the rate of detachment of molecular motors from cytoskeletal
filaments can be biochemically controlled [40–42], drugs that
target the degradation structure fmn g and/or transport velocity v
can have complex interactions with cn -decreasing antivirals,
amplifying their beneficial or harmful effects. Indeed, these
‘‘indirect’’ antiviral drugs (such as Gleevec) that affect transport
have been shown to block vaccinia virus infection [6].
The transport-transformation model can also be used to
potentially explain resensitization of antivirals. Moreover, when
more than one drug is administered, cross resistance can also arise
[43,44]. Within the language of our model, two drugs that increase
the same mn , or decrease the same cn are expected to drive a crossresistant mutation. Although drug-resistant mutants typically have
overall lower replicative efficiency in absence of drug, there is recent
evidence that antiviral protein resistant mutants of the bacteriophage w6174 can carry fitness levels above that of the wild type
[45]. Although the physics of the bacteriophage infection process is
different from the filament-nucleus mechanism described by our
model, the mathematical structure of our transport-based model
does allow for the possibility for the nuclear entry probability of a
drug-resistant mutant to be lower than that of the wild-type. More
refined experiments of efficiencies of reaching intermediate stages
during the infection process may help to refine our genotypical
understanding of drug resistance. Analysis of our model also
potentially provides a guide for probing the nature of the dynamics
within the PN region, based on measurements of the first infection
times after initial entry into the host cell. First passage times between
viral material states, as well as the resulting biophysical implications
are discussed in the Supporting Information.
a decreasing function of degradation rates mn for all n and of
entry rates kn for n=n ;
an increasing function of nuclear import kn ;
an increasing function of the transformation rates cn for
mn §mnz1 such as in Fig. 3(d) and Fig. 3(h);
a non-monotonic function of cn for mn vmnz1 . Here, optimal
infectivity probabilities exist as a function of cn such as in
Figs. 3(a–c) and Fig. 3(a–g);
Supporting Information
Supporting Information file with figures and legends.
Found at: doi:10.1371/journal.pone.0008165.s001 (0.06 MB
PDF)
File S1
Acknowledgments
Maxima in the infection probability Q also imply that apparent
drug resistant strains can arise through mutations in genes
unrelated to the viral components that directly interact with the
antiviral drug. For degradation rates mn that are non-monotonic in
n, the infection probability Q may contain maxima or saddles as a
function of the cn . Antiviral drugs designed to simply inhibit
processes (such as reverse transcription) by reducing the appropriate cn , must be administered as to not increase Q. Moreover,
The authors thank M. Bukrinsky, B. Lee, R. Sun, P. Krogstad, and S.
Chow for important discussions. Part of this work was performed during
the ‘‘Optimal Transport’’ program at the Institute for Pure and Applied
Mathematics (IPAM) at UCLA.
Author Contributions
Analyzed the data: MRD TC. Wrote the paper: MRD TC.
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